{"id":548,"date":"2013-05-12T03:30:33","date_gmt":"2013-05-12T03:30:33","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=548"},"modified":"2025-12-09T12:17:41","modified_gmt":"2025-12-09T18:17:41","slug":"exponents-and-roots","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/exponents-and-roots\/","title":{"rendered":"Evaluating Expressions with Exponents and Roots"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_pH3Rqi7M-I4\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_pH3Rqi7M-I4\" data-source-videoID=\"pH3Rqi7M-I4\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Evaluating Expressions with Exponents and Roots Video\" height=\"720\" width=\"1280\" class=\"size-full\" data-matomo-title = \"Evaluating Expressions with Exponents and Roots\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_pH3Rqi7M-I4:hover {cursor:pointer;} img#videoThumbnailImage_pH3Rqi7M-I4 {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/05\/exponents-thumb.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_pH3Rqi7M-I4\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_pH3Rqi7M-I4\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Evaluating Expressions with Exponents and Roots\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_pH3Rqi7M-I4\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_pH3Rqi7M-I4\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_pH3Rqi7M-I4\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>In this video, I&#8217;m going to talk about exponents and roots. An <strong>exponent<\/strong> is a number that tells you how many times to multiply the base number by itself. For instance, if we have 2 raised to the second power, \\(2^2\\), what this is telling us is that we need to multiply 2 times 2.<\/p>\n<p>Now, if we had 2 raised to the third power,\\(2^3\\), this would be the same as 2 times 2 times 2. The exponent is saying how many times to multiply this number by itself. A <strong>root<\/strong> is kind of the opposite of this. Let&#8217;s take, for instance, 16. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/square-root-of-sixteen-2.png\" alt=\"\" width=\"376\" height=\"184\" class=\"aligncenter size-full wp-image-87547\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/square-root-of-sixteen-2.png 376w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/square-root-of-sixteen-2-300x147.png 300w\" sizes=\"auto, (max-width: 376px) 100vw, 376px\" \/><\/p>\n<p>The radical symbol tells us that we&#8217;re taking a root.<\/p>\n<p>If there is no number in the index spot, the number is assumed to be 2. Without any numbers there, the radical sign designates a square root, or a root to the power of 2. The square root of 16 is a number that has to be multiplied by itself twice in order to equal 16.<\/p>\n<p>The square root of 16, if it is squared or raised to the second power, it equals 16. The square root of 16 is 4. 4 raised to the second power is equal to 16.<\/p>\n<p>Roots and exponents are kind of inverse operations, but you can actually express roots as exponent.<\/p>\n<p>For instance, if we wanted to go back to the square root of 16, we can write the square root of 16 as 16 to the \\(\\frac{1}{2}\\): \\(16^{\\frac{1}{2}}\\)<\/p>\n<p>Essentially, if you have 1 over a number as an exponent, you&#8217;re taking that number&#8217;s root of the number in question. Like I was saying, the 2 that&#8217;s omitted for a square root, that&#8217;s the same as \\(\\frac{1}{2}\\) (the power of \\(\\frac{1}{2}\\)).<\/p>\n<p>You can also have other fractions or other powers of roots. For instance, you might take 8 to the \\(\\frac{1}{3}\\), or third root of 8. This would be a number that you have to multiply by itself three times in order to achieve this. The third power 8 raised to the third power is equal to 8.<\/p>\n<p>This number would be the same as 2, because 2 times 2 times 2, or 2 to the third power, is equal to 8.<\/p>\n<p>A couple other things about exponents. Any number raised to the 1 is just the number itself. 2 raised to the 1 is just 2. Any real number raised to the 0 is equal to 1. It doesn&#8217;t matter if it&#8217;s a positive number or a negative number. Any real non-zero number raised to the power of 0 is equal to 1.<\/p>\n<p>You may notice that all of the exponents and roots we&#8217;ve looked at so far here are whole numbers or fractions of whole numbers.<\/p>\n<p>You can, in fact, have roots that are, for instance, say this is 2 raised to the 2.3. You can have this sort of quantity, but it&#8217;s not going to be easy to calculate by hand, and really you&#8217;re going to use a calculator if you encounter anything that has a decimal in the exponent. <\/p>\n<p>That&#8217;s a good overview of exponents and roots.<\/p>\n<\/div>\n<\/div>\n\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":9,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-548","1":"page","2":"type-page","3":"status-publish","5":"page_category-manipulating-expressions-1","6":"page_category-math-advertising-group","7":"page_category-video-pages-for-study-course-sidebar-ad","8":"page_type-video"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/548","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=548"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/548\/revisions"}],"predecessor-version":[{"id":266164,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/548\/revisions\/266164"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=548"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}